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This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 451. |
Find the differential equation corresponding to the equation `y=Ae^(2x)+Be^(-x)`. |
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Answer» `y=Ae^(2x)+Be^(-x)`……….`(1)` `implies (dy)/(dx)=2Ae^(2x)+Be^(-x)`………`(2)` `implies (d^(2)y)/(dx^(2))=4Ae^(2x)+Be^(-x)`………….`(3)` Adding eqs. `(1)` and `(2)` `(dy)/(dx)+y=3Ae^(2x)` ……………`(4)` Adding eqs. `(2)` and `(3)` `(d^(2)y)/(dx^(2))+(dy)/(dx)=6A*e^(2x)` `implies (d^(2)y)/(dx^(2))+(dy)/(dx)=2((dy)/(dx)+y)` [From eq. `(4)`] `implies (d^(2)y)/(dx^(2))-(dy)/(dx)-2y=0`. |
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| 452. |
Find the order and degree of the differential equation . `(d^(2)y)/(dx)^(2)+x((dy)/(dx))^(3)-1=0` |
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Answer» In this differential equation, the highest derivative is `(d^(2)y)/(dx^(2))`. Therefore the order and degree of this differential equation are `2` and `1` respectively. |
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| 453. |
The slope of a curve at point `(x,y)` is equal to sum of coordinate of that point. Represent it in form of a differential equation. |
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Answer» Required differential equation `(dy)/(dx)=x+y`. |
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| 454. |
Find the differential equation corresponding to the equation `y=A*e^(x)+B`. |
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Answer» `y=A*e^(x)+B`………`(1)` `implies(dy)/(dx)=A*e^(x)`………`(2)` `implies (d^(2)y)/(dx^(2))=A*e^(x)`…………..`(3)` From eqs. `(2)` and `(3)` `(d^(2)y)/(dx^(2))=(dy)/(dx)`. |
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| 455. |
If `A` and `B` are arbitrary constants, then find the differential equation corresponding to `y=Acos(x+B)`. |
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Answer» Given equation is `y=Acos(x+B)`……..`(1)` `implies (dy)/(dx)=-Asin(x+B)`………..`(2)` `implies (d^(2)y)/(dx^(2))=-Acos(x+B)`…………`(3)` The second order derivative is taken because there are two arbitrary constants. From eqs. `(1)` and `(3)` `(d^(2)y)/(dx^(2))=-y`. |
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| 456. |
Solve the differential equation `(1 + x^(2))dy+2xydx = sin^(2) x dx` |
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Answer» `(dy)/(dx) + (2x)/(1+x^(2))y = (sin^(2)x)/(1 + x^(2))` ...(1) Equation (1) is the linear differential equation of the type `(dy)/(dx) + Py = Q`, where `P = (2x)/(1+x^(2))` and `Q = (sin^(2)x)/(1+x^(2))` `:. I.F. = e^(int(2x)/(1+x^(2))dx)` `= e^(ln(1+x^(2)))` `= 1 + x^(2)` Thus, the solution of (1) is `y(1+x^(2)) = int (sin^(2)x)/(1+x^(2))(1+x^(2))dx + c` `= (1)/(2)int(1-cos 2x)dx + c` `rArr y(1+x^(2)) = (1)/(2)(x-(sin 2x)/(2))+c` |
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| 457. |
Solve `(xsiny/x)dy=(ysiny/x-x)dx`. |
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Answer» `(siny/x)(dy)/(dx) = (y/x.siny/x-1)` Put `y=vx` `rArr sinv(v+x(dv)/(dx))=(vsinv-1)` or `sinv(xdv)/(dx) =-1` or `intsinvdv=-int(dx)/(x)` `rArr cosv=log_(e)x+c` `rArr cosy/x=log_(e )x+c` |
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| 458. |
Solve the differential equation `(dy)/(dx)=sec^(2)x`. |
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Answer» `(dy)/(dx)=sec^(2)x` Integrate both sides with respect to `x` `int(dy)/(dx)*-dx=intsec^(2)xdx+c` `implies int dy=intsec^(2)xdx+c` `impliesy=tanx+c`. |
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| 459. |
Solve the differential equation `(dy)/(dx)=(1)/(x)`. |
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Answer» `(dy)/(dx)=(1)/(x)` Integrate both sides with respect to `x` `int(dy)/(dx)*-dx=int(1)/(x)dx+c` `implies intdy=int(1)/(x)dx+c` `implies y=log_(e)x+c`. |
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| 460. |
Find the general solution of the differential equations `(dy)/(dx)=(1-cosx)/(1+cosx)` |
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Answer» Given , `(dy)/(dx)=(1-cosx)/(1+cosx)` `implies dy=(1-cosx)/(1+cosx)dx` `implies intdy=int(1-cosx)/(1+cosx)dx` `implies y=int(2sin^(2)(x//2))/(2 cos^(2)(x//2))dx` `implies y=int tan^(2)(x//2)dx` `implies y=int[sec^(2)((x)/(2))-1]dx` `implies y=int sec^(2)((x)/(2))dx-int 1dx` `implies y=2tan"(x)/(2)-x+C`. |
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| 461. |
Solve the differential equation `(dy)/(dx)=sin(2x+5)`. |
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Answer» `(dy)/(dx)=sin(2x+5)` Integrate both sides with respect to `x` `int (dy)/(dx)=intsin(2x+5)dx+c` `implies indy=intsin(2x+5)dx+c` `implies y=-(1)/(2)cos(2x+5)+c`. |
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| 462. |
Solve the differential equation `(dy)/(dx)=sin^(4)x*cosx`. |
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Answer» `(dy)/(dx)=sin^(4)x*cosx` Integrate both sides with respect to `x` `int(dy)/(dx)*-dx=intsin^(4)x cosx dx+c` Let `sinx=t` `implies intdy=intt^(4)*-dt+-cimpliescosx=-(dt)/(dx)` `cosxdx=dt` `implies y=(1)/(5)t^(5)+c` `implies y=(1)/(5)sin^(5)x+c`. |
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| 463. |
Show that the given differential equation is homogeneous and solve each of them.`(x^2-y^2)dx+2x ydy=0` |
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Answer» `2xydy=(y^(2)-x^(2))dx` `implies (dy)/(dx)=(y^(2)-x^(2))/(2xy)` `implies v+x(dv)/(dx)=(v^(2)x^(2)-x^(2))/(2x^(2)v)` Let `y=vx` `implies (dy)/(dx)=u+(dv)/(dx)` `implies x(dv)/(dx)=(v^(2)-1)/(2v)-v=(-1(1+v^(2)))/(2v)` `implies(2v)/(1+v^(2))dv=-(dx)/(x)` `implies int(2v)/(1+v^(2))dv=-int(dx)/(x)+logc` `implies log(1+v(2))=-logx+logc` `implies 1+v^(2)=(c )/(x)` `impliesx^(2)+v^(2)x^(2)=cx` `implies x^(2)+y^(2)=cx` |
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| 464. |
Solve the differential equation `(1+x^(2))(dy)/(dx)=x`. |
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Answer» `(1+x^(2))(dy)/(dx)=x` Integrate both sides with respect to `x` `(dy)/(dx)=(x)/(1+x^(2))` Integrate both sides with respect to `x` `(dy)/(dx)*-dx=int(x)/(1+x^(2))dx+c` Let `1+x^(2)=t` `implies int dy=int(x)/(1+x^(2))+cimplies2x=(dt)/(dx)` `implies y=int(dt)/(dx)+cimpliesxdx=(dt)/(2)` `implies y=(1)/(2)logt+c` `impliesy=(1)/(2)log(1+x^(2))+c`. |
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| 465. |
If m and n are order and degree of the differential equation `((d^2y)/(dx^2))^5+(4((d^2y)/(dx^2))^3)/((d^3y)/(dx^3))+(d^3y)/(dx^3)=x^2-1`A. m = 3, n = 3B. m = 3, n= 2C. m = 3, n = 5D. m = 3, n = 1 |
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Answer» Correct Answer - B The given differential equation can be written as `((d^(3)y)/(dx^(3)))^(2)-(x^(2)-1)(d^(3)y)/(dx^(3))+((d^(3)y)/(dx^(3)))((d^(2)y)/(dx^(2)))^(5)+4((d^(2)y)/(dx^(2)))^(3)=0` Clearly, its order is 3 and degree is 2. `therefore" m=3 and n=2` |
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| 466. |
Find the solution of the differential equation `cos ydy+cosxsinydx=0`given that `y=pi//2`, when `x=pi//2.` |
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Answer» `cosy*dy+cosxsinydx=0` `implies (cosy)/(siny)dy+cosxdx=0` Inetegrate both sides `int(cosy)/(siny)dy+intcosxdx=0` Let `siny=t` `impliesint(1)/(t)dt+intcosxdx=0` `impliescosydy=dt` `implieslogt+sinx=c` `implies log(siny)+sinx=c`…..`(1)` Given that `x=pi//2` if `y=pi//2` `:. log sin"(pi)/(2)+sin"(pi)/(2)=c` `impliesc=1` `:.` From eq. `(1)` `log(siny)+sinx=1`. |
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| 467. |
Writhe the order of the differential equation ofthe family of circles of radius `rdot`A. 2B. 3C. 4D. none of these |
| Answer» Correct Answer - A | |
| 468. |
The order of the differential equation `((d^(2)y)/(dx^(2)))^(3)=(1+(dy)/(dx))^(1//2)`A. 2B. 3C. `1//2`D. 6 |
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Answer» Correct Answer - A The given differential equation can be written as `((d^(2)y)/(dx^(2)))^(6)=1+(dy)/(dx)` Clearly, its order is 2 and degree is 6. |
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| 469. |
Solve the differential equation `(dy)/(dx)=(1-cos2y)/(1+cos2y)`. |
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Answer» `(dy)/(dx)=(1-cos2y)/(1+cos2y)` `=(2sin^(2)y)/(2cos^(2)y)=tan^(2)y` `implies cot^(2)ydy=dx` `implies intcot^(2)ydy=intdx+c` `implies int(cosec^(2)y-1)dy=intdx+c` `implies-coty-y=x+c`. |
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| 470. |
The solution of the equation `(dy)/(dx)=(x+y)/(x-y)`, isA. `C(x^(2)+y^(2))^(1//2)+e^(tan^(-1)((y)/(x)))=0`B. `C(x^(2)+y^(2))^(1//2)+e^(tan^(-1)((y)/(x)))`C. `C(x^(2)-y^(2))^(1//2)+e^(tan^(-1)((y)/(x)))`D. none of these |
| Answer» Correct Answer - B | |
| 471. |
Write the order of the differential equation whosesolution is `y=acosx+b s in x+c e^(-x)dot`A. 3B. 2C. 1D. none of these |
| Answer» Correct Answer - A | |
| 472. |
Order and degree of the differential equation `(d^(2)y)/(dx^(2))={y+((dy)/(dx))^(2)}^(1//4)` areA. 4 and 2B. 1 and 2C. 1 and 4D. 2 and 4 |
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Answer» Correct Answer - D The given differential equation can be written as `((d^(2)y)/(dx^(2)))^(4)={y+((dy)/(dx))^(2)}` Clearly, its order and degree are 2 and 4 respectively. |
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| 473. |
If `p a n d q`are the order and degree of the differentialequation `y(dy)/(dx)+x^3( d^2y)/(dx^2)+x y=cos x , t h e n``p q`d. none of theseA. p lt qB. p = qC. p gt qD. none of these |
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Answer» Correct Answer - C The given differential equation is of order 2 and degree 1. `therefore" p = 2 and q = 1 "rArr p gt q` |
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| 474. |
find the order and degree of D.E : (1) ` ((d^(2)y)/(dx^(2))^(2)) + ((dy)/(dx)))^(3) = e^(x)` (2) ` sqrt(1 + 1/((dy)/(dx))^(2))= ((d^(2)y)/(dx^(2)))^(3/2)` (3) ` e^((dy)/(dx))+ (dy)/(dx) =x`A. 2, 4B. 3, 4C. 4, 3D. 2, 3 |
| Answer» Correct Answer - D | |
| 475. |
` [(d^(3)y)/(dx^(3)) +x]^(5/2) = (d^(2)y)/(dx^(2))`A. 2, 7B. 3, 7C. 5, 7D. 3, 5 |
| Answer» Correct Answer - D | |
| 476. |
The order and degree of the differential equation `rho=({1+((dy)/(dx))^(2)}^(3//2))/((d^(2)y)/(dx^(2)))` are respectivelyA. 2,2B. 2,3C. 2,1D. none of these |
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Answer» Correct Answer - A The given equation when expressed as a polynomial in derivatives is `rho^(2)((d^(2)y)/(dx^(2)))^(2)={1+((dy)/(dx))^(2)}^(3)` Clearly, it is a second order differential equation of degree 2. |
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| 477. |
The degree of the differential equation `((d^(3)y)/(dx^(3)))^(2//3)+4-3(d^(2)y)/(dx^(2))+5(dy)/(dx)=0,` isA. 1B. 2C. 3D. none of these |
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Answer» Correct Answer - B We have `((d^(3)y)/(dx^(3)))^(2//3)+4-3(d^(2)y)/(dx^(2))+5(dy)/(dx)=0` `rArr" "((d^(3)y)/(dx^(3)))^(2)+(3(d^(2)y)/(dx^(2))-5(dy)/(dx)-4)^(3)` Clearly, it is a differential equation of degree 2. |
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| 478. |
The degree and order of the differential equation` [ 1+((dy)/(dx))^(3)]^(7/3)=7((d^(2)y)/(dx^(2)))` respectively areA. 7, 2B. 2, 3C. 2, 7D. 2, 21 |
| Answer» Correct Answer - B | |
| 479. |
`root3((dy)/(dx))=tanx` . Find order and degree of differential equation.A. `1, (1)/(3)`B. 1, 3C. 1, 1D. 3, 1 |
| Answer» Correct Answer - C | |
| 480. |
The order and degree of D.E. `(d^(2)y)/(dx^(2))=root3(1+((dy)/(dx))^(2))` areA. 2, 3B. 2, 4C. 1, 4D. 4, 1 |
| Answer» Correct Answer - A | |
| 481. |
The solution of differential equation `(dy)/(dx)+(y)/(x)=sin x` isA. `x(y+cos x)=sin x+C`B. `x(y-cos x)=sin x+C`C. `xy cos x=sin x+C`D. `x(y+cosx)=cos +C` |
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Answer» Given differential equation is `(Dy)/(dx+y(1)/(x)=sinx` which is linear differential equation. Here, `P=(1)/(x)and Q=sin x` `therefore IF=e^(int(1)/(x)dx)=e^(logx)=x` The general solution is `y.x.=intx.sin xdx+C` `"Take" I=intx sin xdx` `-x cos x -f -cos xdx` `-x cos x+sin x` Put the value l in Eq. (i), we get `xy=-x cos x+sin x+C` `Rightarrow x(y+cos x)=sin x+C` |
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| 482. |
`(d^(3)y)/(dx^(3))+x((d^(2)y)/(dx^(2)))^(3)+y((dy)/(dx))^(4)=0` . Find order and degree of differential equation.A. 3, 4B. 4, 3C. 3, 1D. 1, 3 |
| Answer» Correct Answer - C | |
| 483. |
`(d^(2)y)/(dx^(2))=[1+(dy)/(dx)]^(3//2)` . Find order and degree of differential equation.A. 2, 1B. 2, 3C. 2, 2D. 3, 2 |
| Answer» Correct Answer - C | |
| 484. |
The solution of differential equation `(dy)/(dx)=e^(x-y)+x^(2)e^(-y)`isA. `y=e^(x-y)-x^(2)e^(-y)+C`B. `e^(y)-e^(x)=(x^(3))/(3)+C`C. `e^(x)+e^(y)=(x^(3))/(3)+C`D. `e^(x)-e^(y)=(x^(3))/(3)+C` |
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Answer» Given that, `(dy)/(dx)=e^(x-y)+x^(2)e^(-y)` `Rightarrow (dy)/(dx)=e^(x)e^(-y)+x^(2)e^(-y)` `Rightarrow (dy)/(dx)=(ex^(2)+x^(2))/(e^(y))` On integrating both sides, we get `inte^(y) dy=int(e^(x)+x^(2))dx` `Rightarrow e^(y)=e^(x)+(x^(3))/(3)+C``Rightarrow e^(y)-e^(x)=(x^(3))/(3)+C` |
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| 485. |
`(y+(dy)/(dx))^(2)+x(dy)/(dx)=x^(2)` . Find order and degree of differential equation.A. 1, 2B. 2, 1C. 1, 1D. 2, 2 |
| Answer» Correct Answer - A | |
| 486. |
`(d^(4)y)/(dx^(4))=[1+((dy)/(dx))^(2)]^(3//2)` . Find order and degree of differential equation.A. 4, 6B. 4, 3C. 2, 4D. 4, 2 |
| Answer» Correct Answer - D | |
| 487. |
The solution of differential equation `(dy)/(dx)+(2xy)/(1+x^(2))=(1)/(1+x^(2))^(2)"is"`A. `y(1+x^(2))=C+tan^(-1)x`B. `(y)/(1+x^(2))=C+tan^(-1)x`C. `y log (1+x^(2))=C+tan^(-1)x`D. `(1+x^(2))=C+sin^(-1)x` |
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Answer» Given that, `" "(dy)/(dx)=(2xy)/(1+x^(2))=(1)/((1+x^(2))^(2))` Here, `" "P=(2x)/(1+x^(2)) and Q = (1)/((1+x^(2))^(2))` which is a linear differential equation. `therefore " "IF=e^(int(2x)/(1+x^(2))dx)` Put `" "1+x^(2)=trArr2xdx=dt` `therefore " "IF=e^(int(dt)/(t))=e^(logt)=e^(log(1+x^(2)))=1+x^(2)` The general solution is `" "y*(1+x^(2))=int(1+x^(2))(1)/((1+x^(2))^(2))+C` `rArr" "y(1+x^(2))=int(1)/(1+x^(2))dx+C` `rArr" "y(1+x^(2))=tan^(-1)x+C` |
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| 488. |
`y=(dy)/(dx)+sqrt(1+(dy)/(dx))` . Find order and degree of differential equation.A. 1, `(1)/(2)`B. 2, 1C. 1, 2D. 1, 1 |
| Answer» Correct Answer - C | |
| 489. |
`|(1," "y),(1,x(dy)/(dx)+sqrt(1+((dy)/(dx))^2))|-y=0` . Find order and degree of differential equation.A. 2, 1B. 1, 2C. 1, 1D. 1, 4 |
| Answer» Correct Answer - B | |
| 490. |
(i) The degree of the differential equation `(d^(2)y)/(dx^(2))+e^(dy//dx)=0` is… (ii) The degree of the differential equation `sqrt(1+((dy)/(dx))^(2))="x is".......` (iii) The number of arbitrary constant in the general solution of differential equation of order three is.. (iv) `(dy)/(dx)+(y)/(x log x)=(1)/(x)` is an equation of the type.... (v) General solution of the differential equation of the type is givven by... (vi) The solution of the differential `(xdy)/(dx)+2y=x^(2)` is.... (vii) The solution of `(1+x^(2))(dy)/(dx)+2xy-4xy^(2)=0` is... (viii) The solution of the differential equation `ydx+(x+y)dy=0` is .... (ix) Genergal solution of `(dy)/dx)+y=sin x` is.... (x) The solution of differential equation cot y `dx=xdy ` is.... (xi) The integrating factor of `(dy)/(dx)+y=(1+y)/(x)` is..... |
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Answer» (i) Given differential equation is `" "(d^(2)y)/(dx^(2))+e^((dy)/(dx))=0` Degree of this equation is not defined. (ii) Given differential equation is `sqrt(1+((dy)/(dx))^(2))=x` So, degree of this equation is two. (iii) There are three arbitary constists. (iv) Given differential equation is `(dy)/(dx)+(y)/(xlogx)=(1)/(x)` The equation is of the type `(dy)/(dx)+Py=Q` (v) Given differential equation is `" "(dx)/(dy)+P_(1)x=Q_(1)` The equation of the form `" "x*IF=intQ(IF)dy+C" "i.e., xe^(int(Pdy))=intQ{e^(intPdy)}dy+C` (vi) Given differential equation is `" "x(dy)/(dx)+2y=x^(2)rArr(dy)/(dx)+(2y)/(x)=x` This equation of the form `(dy)/(dx)+Py=Q`. `therefore" "IF=e^(int(2)/(x)dx)=e^(2logx)=x^(2)` The general solution is `" "yx^(2)=intx*x^(2)dx+C` `rArr" "yx^(2)=(x^(4))/(4)+C` `rArr" "y=(x^(2))/(4)+Cx^(-2)` (vii) Given differential equations is `" "(1+x^(2))(dy)/(dx)+2xy-4x^(2)=0` `rArr" "(dy)/(dx)+(2xy)/(1+x^(2))-(4x^(2))/(1+x^(2))=0` `rArr" "(dy)/(dx)+(2x)/(1+x^(2))y=(4x^(2))/(1+x^(2)` `therefore" "IF=e^(int(2x)/(1+x^(2))dx)` Put `" "1+x^(2)=trArr2xdx=dt` `therefore " "IF=e^(int(dt)/(t))=e^(logt)=e^(log(1+x^(2)))=1+x^(2)` The general solution is `" "y*(1+x^(2))=int(1+x^(2))(4x^(2))/((1+x^(2)))dx+C` `rArr" "(1+x^(2))y=int4x^(2)dx+C` `rArr" "(1+x^(2))y+4(x^(3))/(3)+C` `rArr" "y=(4x^(3))/(3(1+x^(2)))+C(1+x^(2))^(-1)` Given differential equation is `rArr" "ydx+(x+xy)dy=0` `rArr " "ydx+x(1+y)dy=0 ` `rArr" "(dx)/(-x)=((1+y)/(y))dy` `rArr" "int(1)/(x)dx=-int((1)/(y)+1)dy" "` [on integrating] `rArr" "log(x)=-log(y)-y+logA` `" "log(xy)+y=logA` `rArr" " xye^(y)=A` `rArr" "xy=Ae^(-y)` (ix) Given differential equation is `" "y*e^(x)=inte^(x)sinxdx+C` Let `" "I=inte^(x)sinxdx" "...(i)` `" "I=sinxe^(x)-intcosxe^(x)dx` `" "=sinxe^(x)-cosxe^(x)+int(-sinx)e^(x)dx` `" "2I=e^(x)(sinx-cosx)` `" "I=(1)/(2)e^(x)(sinx-cosx)` From Eq. (i) `" "y*e^(x)=(x)/(2)(sinx-cosx)+C` `rArr" "y=(1)/(2)(sinx-cosx)+C*e^(-x)` (x) Given differential equation is `" "cotydx=xdy` `rArr" "(1)/(x)dx=tanydy` On intergrating both sides, we get `rArr" "int(1)/(x)dx=inttanydy` `rArr" "log(x)=log(sec y)+logC` `rArr" "log((x)/(secy))=logC` `rArr" "(x)/(secy)=C` `rArr" "x=Csecy` (xi) Given differential equation is `" "(dy)/(dx)+y=(1+y)/(x)` `" "(dy)/(dx)+y=(1)/(x)+(y)/(x)` `rArr" "(dy)/(dx)+y(1-(1)/(x))=(1)/(x)` `therefore" "IF=e^(int(1-(1)/(x))dx)` `" "=e^(x)*e^(-logx)=(e^(x))/(x)` |
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| 491. |
The degree of the differential equation `((d^(2)y)/(dx^(2)))+((dy)/(dx))^(2)=x sin((d^(2)y)/(dx))`, isA. 1B. 2C. 3D. none of these |
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Answer» Correct Answer - D Clearly, the given differential equation is not a polynomia in differential coefficients. So, its degree is not defined. |
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| 492. |
The degree of the differential equation `(d^(2)y)/(dx^(2))+3((dy)/(dx))^(2)=x^(2)log((d^(2)y)/(dx^(2))),` isA. 1B. 2C. 3D. none of these |
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Answer» Correct Answer - D Since the equation is not a polynomial in all differential coefficients. So, its degree is not defined. |
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| 493. |
Let thepopulation of rabbits surviving at a time t be governed by the differentialequation `(d p(t)/(dt)=1/2p(t)-200.`If `p(0)""=""100`, then p(t) equals(1) `400-300""e^(t//2)`(2) `300-200""e^(-t//2)`(3) `600-500""e^(t//2)`(4) `400-300""e^(-t//2)`A. `40-300e^(t//2)`B. `200-200e^(-t//2)`C. `600-500e^(t//2)`D. `400-300e^(-t//2)` |
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Answer» Correct Answer - A `(dp)/(dt) = (p-400)/(2)` `rArr (dp)/(p-400)=1/2dt` Integrating, we get `"ln "|p-400|=1/2t+c` When `t=0, p=100`, we have ln 300=c `therefore "ln"|(p-400)/(300)|=t/2` `rArr |p-400|=300e^(t//2)` `rArr 400-p=300e^(t//2)` (as `p lt 400)` `rArr p=400-300e^(t//2)` |
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| 494. |
Let thepopulation of rabbits surviving at a time t be governed by the differentialequation `(d p(t)/(dt)=1/2p(t)-200.`If `p(0)""=""100`, then p(t) equals(1) `400-300""e^(t//2)`(2) `300-200""e^(-t//2)`(3) `600-500""e^(t//2)`(4) `400-300""e^(-t//2)`A. (a) `400-300e^(t/2)`B. (b) `300-200e^(t/2)`C. (c) `600-500e^(t/2)`D. (d) `400-300e^(t/2)` |
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Answer» Correct Answer - (a) Given, differential equation is `(dp)/dt-1/2p(t)=-200` a linear differential equation. Here, `p(t)=(-1)/2,Q(t)=-200` `IF=e^(int-(1/2)dt)=e^(t/2)` Hence, solution is `p(t) cdot IF = int Q(t) cdot IF dt` `p(t) cdot e^(t/2)=int-200 cdot e ^(t/2)dt` `p(t) cdot e^(t/2)=400 cdot e ^(t/2)+k` `rArr p(t)=400 +ke^(-1//2)` If p(0)=100, then k=-300 `rArr p(t)=400-300e^(1/2)` |
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| 495. |
Let thepopulation of rabbits surviving at a time t be governed by the differentialequation `(d p(t)/(dt)=1/2p(t)-200.`If `p(0)""=""100`, then p(t) equals(A. `600-500e^(t//2)`B. `400-300e^(-t//2)`C. `400-300e^(t//2)`D. `300-200e^(-t//2)` |
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Answer» Correct Answer - C We have, `(d)/(dt)(f(t))=(1)/(2)p(t)-200` `rArr" "(d)/(dt)(p(t))+(-(1)/(2))p(t)=-200" …(i)"` This is a linear differential equation with I.F. `=e^(int-(1)/(2)dt)=e^(-(1)/(2))` Multiplying both sides of (i) by I.F. `=e^(-t//2)`, we obtain `e^(-t//2)(d)/(dt)(p(t))+(-(1)/(2))p(t)e^(-t//2)=-200e^(-t//2)` Integrating both sides with respect to t, we get `P(t)e^(-t//2)=400e^(-t//2)+C" ...(ii)"` Putting t = 0 and p(0) = 100, we get `100=400+CrArr C=-300` Putting C =` -300`, we get `p(t)e^(-t//2)=400e^(-t//2)-300` `rArr" "p(t)=400-300e^(t//2)` |
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| 496. |
The integrating factor of differential equation `dy/dx+y=(1+y)/x` isA. `(x)/(e^(x))`B. `(e^(x))/(x)`C. `xe^(x)`D. `e^(x)` |
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Answer» Correct Answer - B We have, `(dy)/(dx)+(1-(1)/(x))y-(1)/(x)`. `therefore" I.F. "=e^(int(1-(1)/(x))dx)=e^(x-logx)=(1)/(x)e^(x)` |
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| 497. |
The solution of differential equation cosx. Sin ydx + sin x. cosydy = 0 isA. `(sin x)/(sin y) = c`B. ` sinx.siny = cC. sinx + sin y = cD. cos x. cos y = c |
| Answer» Correct Answer - B | |
| 498. |
Let `f:[0,1] rarr R` be such that `f(xy)=f(x).f(y),` for all `x,y in [0,1]` and `f(0) ne 0.` If `y=y(x)` satisfies the differential equation, `dy/dx=f(x)` with `y(0)=1,` then `y(1/4)+y(3/4)` is equql to(A)`5 `(B) `3`(C) `2`(D) `4`A. (a) 5B. (b) 3C. (c) 2D. (d) 4 |
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Answer» Correct Answer - (b) Given , `f(xy)=f(x)cdot f(y),AAx,yin[0,1] …(i)` Putting `x = y = 0` in Eq. (i), we get `f(0) = f(0)cdot f(0)` `rArr f(0)[f(0)-1]=0` `rArr f(0)=1as f(0)ne 0` Now, put y=0 in Eq. (i) , we get `f(0)=f(x)cdot f(0)` `rArr f(x)=1` so, `dy/dx=f(x)rArr dy/dx=1` `rArr int dy=intdx` `rArr y=x+ C` `therefore y(0)=1` `therefore 1=0+C` `rArr C=1` `therefore y=x+1` Now, `y(1/4)=1/4+1 =5/4 and y(3/4)=3/4+1=7/4` `rArr y(1/4)+y(3/4)=5/4+7/4=3` |
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| 499. |
The solution of the differential equation `x(dy)/(dx)+y=x^(3)y^(6)`, isA. `x^(7)=5y^(5)+Cx^(2)y^(5)`B. `2x^(7)=5y^(5)+Cx^(2)y^(5)`C. `5x^(7)=2y^(5)+Cx^(2)y^(5)`D. `2x^(7)=5y^(5)+Cx^(5)y^(2)` |
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Answer» Correct Answer - B The given differential equation can be written as `(1)/(y^(6))(dy)/(dx)+(1)/(xy^(5))=x^(2)` Let `y^(-5)=v`, Then, `-5y^(-6)(dy)/(dx)=(dv)/(dx)rArry^(-6)(dy)/(dx)=-(1)/(2)(dv)/(dx)` Substituting there values in the given differential equation, we get `-(1)/(5)(dv)/(dx)+(1)/(x)v=x^(2)` `rArr" "(dv)/(dx)-(5)/(x)v=-5x^(2)" ...(i)"` This this the standard form of the linear differential equation having integrating factor `"I.F"=e^(int-(5)/(x)dx)=e^(-5logx)=(1)/(x^(5))` Multiplying both sides of (i) by I.F. and integrating w.r.t. x, we get `v-(1)/(x^(5))=int-5x^(2).(1)/(x^(5))dx` `rArr" "(v)/(x^(5))=(5)/(2)x^(-2)+C` `rArr" "y^(-5)x^(5)=(5)/(2)x^(-2)+C,` which is the required solution. |
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| 500. |
The solution of diferential equation `(dy)/(dx) = (1+y^(2))/(1+x^(2))` isA. `y = tan^(-1) x+c`B. `tan^(-1)y = x + c`C. `(y-x) = c(1+xy)`D. tan xy = c |
| Answer» Correct Answer - C | |